Yes, it is a general case that indefinite integrals cannot be expressed in terms of some finite combination of analytical and special functions. That is a more mathematically correct expression of the thing you obviously have in mind when writing "does not exist".

Even more, most of indefinite integrals have this property, and only smaller part of them can be expressed in terms of analytical and special functions. It is also common that the indefinite integral "does not exist" (using your expression), while the definite one does.